The former Manchester United goalkeeper did superbly well to keep out Kirk Broadfoot's header in a match of few chances.
And his vital stop meant the Robins moved within two points of safety despite falling to the foot of the league while Blackpool are still without a goal or a win since Paul Ince's return.
It was the home side that had the first real opportunity of a quiet start in the 17th minute when Matt Phillips' cross found Matt Derbyshire but Liam Fontaine did enough to stop the former Blackburn man from testing Heaton.
The Robins, buoyed by relegation rivals Wolves' draw on Friday night, then had three half-chances in quick succession blocked by Blackpool defenders with Marvin Elliott, Lewin Nyatanga and Jon Stead all trying their luck from distance.
Stead's goalscoring ability helped City record a vital victory against Barnsley last Saturday and he was next to threaten with a looping header from the centre of the penalty area on the half hour mark which was comfortable for Matt Gilks.
Paul Anderson was next to threaten in a half of few clear-cut chances, but the former Liverpool winger fired just over the crossbar after being found by Stephen Pearson.
The home side, aiming to take their unbeaten run in Championship matches against the Robins to five, had half-chances of their own before the break with Tom Ince warming the gloves of Heaton and Derbyshire volleying wide from 12 yards on the stroke of half-time.
Ambitious efforts from Greg Cunningham and Angel Martinez were a long way from testing either goalkeeper at the start of the second 45 as Blackpool's drought without a goal looked set to continue.
It was City doing all the pushing as they looked for a crucial three points and, after heading straight at Gilks, Steven Davies had the ball in the net but the former Derby man was flagged for offside.
Manager Ince introduced Nathan Delfouneso and the Aston Villa loanee was agonisingly close to connecting with Alex Baptiste's ball just two minutes after replacing Neal Eardley.
A minute later they carved out the best chance of the game for Broadfoot but Heaton was up to the challenge and the former Cardiff stopper did superbly well to keep out the Scot's header from Ince's corner.
O'Driscoll responded by introducing Ghana winger Albert Adomah with eight minutes remaining but the ex-Barnet man could not light up a forgettable 90 minutes as the Robins drew 0-0 for the first time this season.